What is the Integrating Factor for x^n*y^m?

In summary, To solve a differential equation with an integrating factor of the form x^n * y^m, you must first multiply the equation by that integrating factor and then find the values of m and n by equating the coefficients of the same powers in the resulting equation.
  • #1
bemigh
30
0
Hey everyone,
I need to find an integrating factor of the form x^n*y^m, to solve a differential equation i have... however i do not know the process to solve for an integration of this form.. .any help??
Thanks
Steph
 
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  • #2
can you give an example problem that your working on?
 
  • #3
the problem is ( 12 + 5xy )dx + (6 (x/y)+ 3x^2)dy =0
and it says, find an integrating factor of the form (x^n) * (y^m), and solve the equation...
thanks sweetie
steph
 
  • #4
RULE 1: Mathematics problems are not solved by staring at a problem until you remember the answer! They are solved by plugging things in and doing the algebra.
So TRY!

If you multiply the equation by [itex]x^ny^m[/itex] you get
[itex](12x^ny^m+ 5x^{n+1}y^{m+1})dx+ (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)dy= 0[/itex]

In order for that to be an exact equation, you must have
[itex](12x^ny^m+ 5x^{n+1}y^{m+1})_y= (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x[/itex]

Do the derivatives and see what m and n must be for those to be equal!

[itex](12x^ny^m+ 5x^{n+1}y^{m+1})_y= 12mx^ny^{m-1}+ 5(m+1)x^{n+1}y^m[/itex]
[itex](6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x= 6(n+1)x^ny^{m-1}+3(n+2)x^{n+1}y^m[/itex]
Coefficients of the same powers must be equal. That gives two equations for m and n.
 
Last edited by a moderator:

1. What is an integrating factor?

An integrating factor is a function that is multiplied by both sides of a differential equation in order to make it easier to solve. It essentially "integrates" the equation, hence the name.

2. How do I determine the integrating factor for a given equation?

The integrating factor for an equation of the form x^n*y^m can be determined by using the formula: IF = e^(n*ln|x| + m*ln|y|). Note that the absolute value signs are important to include.

3. What is the purpose of using an integrating factor?

Using an integrating factor can help to simplify the process of solving a differential equation. It can also allow for the use of different methods, such as separation of variables, to solve the equation.

4. Can any equation be solved using an integrating factor?

No, not all differential equations can be solved using an integrating factor. Some equations may require different methods or may not have a closed-form solution.

5. Are there any limitations to using an integrating factor?

One limitation of using an integrating factor is that it may not always lead to a closed-form solution. Additionally, the process of determining the integrating factor and solving the equation can be time-consuming and tedious.

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